Let ℒ be the class of discrete groups G which admit a finite complex X as classifying space. On ℒ we may define the Euler characteristic χ(G) = χ(X); that this is independent of X follows from the homological interpretation of characteristics.

In the normal procedure for establishing linearized stability criteria a mean steady state which we denote by U is perturbed by a small additional disturbance u in the Eulerian sense. In the linearization it is usual to regard the disturbance as resolved into Fourier components which are additive so far as the first order goes. The linearized problem will then determine a time amplification factor eiσr so that the linearized disturbance is of the form uexp{i(αx + σt)} and the whole motion U + uexp{i(αx + σt)}. As usual it is implied that the real part of complex functions such as uexp{i(αx + σt)} be taken. In cases of instability (imaginary part of σ negative), where the magnitude of the disturbance increases with the time, we are prompted to ask how energy is supplied to the disturbance. This question is especially relevant to many of the classical problems of fluid motion instability in which the systems are self-contained in the sense that energy is neither supplied nor withdrawn by an outside agency. If such a system is non-dissipative we expect that the total energy of the organized motion and the electromagnetic field, if present, shall be conserved, whilst such energy in a dissipative system will be gradually reduced by heat losses. In either case energy which is taken into a growing disturbance must be provided from the mean energy stored in the system in its initial state. One of the simplest cases of instability to examine from this point of view is the Helmholtz instability of a plane in viscid vortex sheet. This system is self-contained and conservative in the energy of the organized motion, and moreover the only form of energy with which we have to deal is kinetic energy. In the first part of this note we examine how disturbance kinetic energy is provided in this case by the break up of the organized motion of the mean stream. In the second part we consider what happens to the energy balance of a current-vortex sheet in the non-dissipative case. There are here two forms of energy, kinetic and magnetic, and it is shown that when the interface is unstable the overall magnetic energy increases at the expense of the kinetic energy.

The basic equations of non-relativistic magnetohydrodynamics are briefly reviewed. Beginning with the energy-momentum tensor for the coupled electromagnetic and fluid fields relativistic energy and momentum equations are formulated. The effect of heat conduction is included and relativistic constitutive equations are discussed. A relativistic first law of thermodynamics is developed in terms of internal energy and entropy. Finally, the second law and entropy production rate are derived in a covariant fashion.

Malecot(4) under certain conditions derived the formula for the covariance of the genotypic values of a quantitative character in two individuals AI and AII, which were related but not by direct descent. This generalized some results of Fisher (l). Kempthorne (2) extended the theory to multiple allelic systems with any degree of epistacy (i.e. interlocular genie interaction) but without linkage. He gave the formula Here is the item in the population variance which can be attributed to the interaction of additive gene effects at r loci and dominance gene effects at s loci. φ and φ′ are the coefficients of relation between the two individuals. The various assumptions normally included under random mating equilibrium were made, i.e. no selection, mutations, maternal effects or differential viability. Kempthorne (2), (3) gave two rather different proofs of this important result. His second proof was the more straightforward, but it was somewhat condensed.

The recurring controversy about clocks and relativity reveals the need for an examination of the meaning and methods of time measurement.

A train of surface waves (water waves under gravity) is normally incident on a cylinder with horizontal generators fixed near the free surface, and is partially transmitted and partially reflected. At a great distance behind the cylinder the wave motion tends to a regular wave train travelling towards infinity; the ratio of its amplitude to the amplitude of the incident wave is the transmission coefficient . The transmission coefficient is studied when the wavelength is short compared to the dimensions of the body; physically (though not for engineering applications) this is the most interesting range of wavelengths, which corresponds to the range of shadow formation and ray propagation in optics and acoustics. The waves are then confined to a thin layer near the free surface, and the transmission under a partially immersed obstacle is then small. In the calculation the boundary condition at the free surface is linearized, viscosity is neglected, and the motion is assumed to be irrotational.

At present the transmission coefficient is known only for a few configurations, all of them relating to infinitely thin plane barriers. A method is now given which is applicable to cylinders of finite cross-section and which is worked out in detail for a half-immersed cylinder of circular cross-section. The solution of the problem is made to depend on the solution of an integral equation which is solved by iteration. Only the first two terms can be obtained with any accuracy, and it appears at first that this is not sufficient to give the leading term in the transmission coefficient at short wavelengths; this difficulty is characteristic of transmission problems. By various mathematical devices which throw light on the mechanism of wave transmission, it is, nevertheless, found possible to prove that the transmission coefficient for waves of short wavelength λ and period 2π/ω incident on a half-immersed circular cylinder of radius a is asymptotically given by

when N = 2πα/λ = ω2α/g is large. Earlier evidence had pointed towards an exponential law. It is suggested that transmission coefficients of order N−4 are typical for obstacles having vertical tangents and finite non-zero radius of curvature at the points where they meet the horizontal mean free surface. For obstacles having both front and rear face plane vertical to a depth a, is probably of order e−2N approximately; if only one of the two faces is plane vertical, is probably of order e−N approximately. Thus is seen to depend critically on the details of the cross-section.

Martin(1) has shown that if the pressure p and the Lagrange variable ψ (ψ = constant on a trajectory) are taken as independent variables, then problems in one-dimensional unsteady flow of an ideal gas reduce to the question of finding solutions of the Monge-Ampère equation

where τ = 1/ρ, ρ being the density, and S = S(ψ) is the specific entropy. Thus the right-hand term is a function of p and ψ depending upon the equation of state and the function S(ψ).

THEOREM 1. If av > 0 and is a convergent series, then

The following well-known theorem has been proved by Iwasawa(1) and can also be deduced without much difficulty from earlier work of Levi(2), Magnus and Witt.

A simple method for constructing irreducible representations of the symmetric group is given. It is particularly suitable for use in quantum mechanical calculation of interactions in many particle systems, using electronic digital computers.

Let f(z)=u(x, y)+iv(x, y) be regular in a domain containing the unit circle C. If where r < 1, then is regular in |z| ≤ 1 and we have and taking real parts and imaginary parts we obtain Poisson's formulae for the harmonic functions u and v.

An account is given of Eshelby's point-force method for solving elastic inclusion problems, and of his equations relating an in homogeneity to its equivalent inclusion. The introduction of complex variable formalism enables explicit solutions to be found in various two-dimensional cases. Strain energies are calculated. The equilibrium shape of an elliptic inclusion exhibits an interesting feature not previously expected. A fresh analysis of stress magnification effects is developed.

A doubly-stochastic matrix is an n × n matrix with non-negative elements such that each row and each column sums to 1. A permutation matrix is the result of permuting the rows of the unit n × n matrix. Birkhoff's theorem states that the doubly-stochastic matrices constitute the convex hull of the permutation matrices. Using Birkhoff's theorem, Farahat and Mirsky (1) showed that each doubly-stochastic matrix could be expressed as a convex combination of n2 − 2n + 2 permutation matrices, though not in general of fewer. Given Birkhoff's theorem, the Farahat-Mirsky refinement can also be proved quite shortly as follows.

It is a classical and well-established result that a unirational algebraic variety of dimension ≪ 2 is necessarily rational. It is also generally agreed, though not so well established, that this result is no longer true for varieties of dimension three. In this connexion, the critical example has been the Enriques threefold E, asserted by Roth (1) to be unirational but not rational. The purpose of this note is to point out a fallacy in the proof of the non-rationality of E and, incidentally, to resolve a difficulty raised by Serre (2) in which E would appear to feature as a counter-example to a well-established general theorem. As far as the author is aware, there are no other examples of nonrational unirational threefolds, so that the question of their existence is still open.

Several forms of the lower-bound variational method for the calculation of the eigenvalues in a wave-mechanical problem are considered, and compared; the particular case of the harmonic oscillator being chosen. All forms have certain unsatisfactory features, but some of them are considerably worse than others. One reason why calculations of lower bounds are in general less satisfactory than Ritz-type calculations of an upper bound is shown to be that whereas, in the presence of a scale factor, this latter wave-function satisfies the virial theorem, in none of the lower-bound wave-functions is this true. Similar calculations are reported for the ground state of the helium atom.

The problem considered in this note arises in the theory of perfect crystal dislocations and I am indebted to Prof. F. R. N. Nabarro for drawing my attention to it.

In a recent paper (3) Mordell proved the following theorem.

Theorem 1. Let xn + i = xi ≥ 0 for all i and x1 + … + xn = 1. Then

The calculation of the interaction between a point multipole moment and an external charge distribution is considered in elliptic coordinates. It is complicated by the singularity at the point moment. The results obtained are useful in calculating hyperfine interactions in diatomic molecules where the elliptic coordinates are the most natural system of coordinates. The results lead to an extension of Heine's expansion of (λ + μ)−1.